On the average internal path length of m-ary search trees

Abstract

Consider an m-ary tree constructed from a random permutation of size n. When all permutations are equally likely, the average internal path length, which may be considered as a cost measure for searching the tree, is shown to be (n+1)H n /(H m −1)+cn+O(nβ),β<1, with $$c = c(m) = - m/(m - 1) - (H_m - 1)^{ - 1} + A_1^{(m)} $$ , where Hkis the kth harmonic number and A 1 (m) is a coefficient obtained by solving a linear system of equations. This result tells us that the average cost of searching unbalanced m-ary trees is essentially the same as that of searching other popular variants of m-ary trees like B-trees and B+-trees where sophisticated methods are used for balancing.